Density Forecast Evaluation in Unstable Environments 1
GLORIA GONZALEZ-RIVERA
Department of EconomicsUniversity of California, Riverside, CA, USA
YINGYING SUN
Bank of the WestSan Francisco, CA, USA
April 18, 2016
1We are grateful to the participants at the International Symposium in Forecasting, Rotterdam (Hol-land), and to the seminar participants at University of Southern California, University of California,San Diego, Universidad Carlos III de Madrid (UC3M), Universidad Autonoma de Madrid, UniversidadComplutense de Madrid, and Bank of Spain for useful comments. Gloria Gonzalez-Rivera acknowledgesthe financial support of the 2015 Chair of Excellence UC3M/Banco de Santander as well as the supportof the Spanish Ministerio de Economıa y Competitividad (grant ECO2012-32854) and the UC-RiversideAcademic Senate grants.
Abstract
We propose a density forecast evaluation method in the presence of instabilities, which are definedas breaks in any conditional moment of interest and/or in the functional form of the conditionaldensity of the process. Within the framework of the autocontour-based tests proposed in Gonzalez-Rivera et al. (2011, 2015), we construct Sup- and Ave-type tests calculated over a collection ofsubsamples in the evaluation period. These tests enjoy asymptotic distributions that are nuisance-parameter free, they are correctly sized and very powerful on detecting breaks in the parametersof the conditional mean and conditional variance. A power comparison with the tests of Rossiand Sekhposyan (2013) shows that our tests are more powerful across the models considered. Weanalyze the stability of a dynamic Phillips curve and find that the best one-step-ahead densityforecast of changes in inflation is generated by a Markov switching model that allows state shiftsin the mean and variance of inflation changes as well as in the coefficient that links inflation andunemployment.
Key Words: Generalized Autocontour-based Testing, Structural breaks, Phillips Curve.
JEL Classification: C01, C22, C53.
1 Introduction
Generally, instability is understood as changes in the parameters of a proposed forecasting model
over the forecasting horizon. For clarification purposes, consider a simple model yt+1 = β′xt+σεt+1
with εt ∼ i.i.d.N(0, 1). The model is unstable over time if the slope coefficients β can change over
the forecasting sample, either smoothly or abruptly to contain one or multiple breaks. We may
also entertain a time varying variance such that σ may be also subject to breaks, and we may
have different conditional probability density functions, e.g. more or less thick tails, over different
periods of time. This definition is general enough to account for most types of instability discussed
in the current applied econometric literature. Up to today, the most comprehensive survey in the
subject is Rossi (2014) in the Handbook of Economic Forecasting that reports extensive empirical
evidence of instabilities in macroeconomic and financial data. Some examples follow.
The instability of predictive regressions, in which the significance of predictive regressors varies
over different subsamples, has been documented in studies of predictability of stock returns (see
Goyal and Welch, 2003; Paye and Timmermann, 2006; Rapach and Zhou, 2014), in exchange rate
predictions (see Rossi, 2006; Rogoff and Stavrakeva, 2008) and in output growth and inflation
forecasts (see Stock and Watson, 2003; Rossi and Sekhposyan, 2010). Naturally linked to this
evidence is the econometric question on testing for parameter stability and structural breaks in
the data, which has an illustrious history. From Chow (1960) test to most recent works such as
Andrews (1993), Andrews and Ploberger (1994) , Pesaran and Timmermann (2002), among others,
testing for breaks has mainly focused on the behavior of the conditional mean. Our contribution
aims to extend testing for instabilities to the full conditional density forecast that includes not only
any conditional measure of interest, e.g. mean, variance, duration, etc. but also the functional form
of the assumed conditional density function. To our knowledge, the literature on this question is
very thin. We only know of the work by Rossi and Sekhposyan (2013) who are also testing for the
instability of the density model but with a different methodology from ours. In the forthcoming
1
sections, we will explain the differences and will provide a comparison between both approaches.
The testing methodology that we propose is based on the AutoContouR (ACR) device introduced
by Gonzalez-Rivera et al. (2011, 2012) and generalized later on in Gonzalez-Rivera and Sun
(2015). In these works, the null hypothesis of our tests is a correctly specified density forecast
(joint hypothesis of correct dynamics in the moments of interest and correct functional form of
the density). Following Diebold’s (1998) work, we calculate Rosenblatt (1952) probability integral
transforms (PIT) associated with the point forecasts. Under the null, the PITs must be i.i.d
uniformly distributed U[0,1]. The Generalized AutoContouR (G-ACR) is a device (set of points)
that is very sensitive to departures from the null in either direction and consequently, it provides
the basis for very powerful tests. More specifically, for a time series of PITs, we construct the
G-ACRs as squares (in the univariate case) of different probability areas within the maximum
square (area of 1) or as hyper-cubes (in the multivariate case) of different probability volumes
within the maximum hyper-cube formed by a multidimensional uniform density [0, 1]n for n ≥ 2.
By statistically comparing the location of the empirical PITs and the volume of the empirical G-
ACRs with the location and volume of the population G-ACRs, we are able to construct a variety of
tests for correct density forecast. Since the shapes of the G-ACRs can be visualized, we can extract
information about where and how the rejection of the null hypothesis comes from. This testing
framework is the foundation for the new stability tests, Sup- and Ave-type statistics, proposed in
this manuscript. In a potentially unstable data environment, we will form rolling subsamples within
the forecasting sample. For every subsample, we apply a battery of G-ACR tests, and to detect
instabilities, we construct a Sup- and an Ave-type statistics. Though the limiting distribution
of these tests is a function of Brownian motions, the tests are nuisance-parameter free and their
distribution can be tabulated.1
The paper is organized as follows. In section 2, we review the G-ACR approach to make the
1Though we focus on out-of-sample density forecasts, the methodology proposed in this paper can also be appliedto in-sample specification testing.
2
exposition self-contained and we introduce the new statistics with their asymptotic distributions. In
section 3, we assess the finite sample properties (size and power) of the tests. We offer an extensive
assessment by considering (i) fixed, rolling, and recursive estimation schemes; (ii) different ratios
of prediction sample size to estimation sample size; and (iii) break points that occur in different
periods of the prediction sample. We also present a comparison of power of our tests with those of
Rossi and Sekhposyan (2013). In section 4, we apply the tests to assess the stability of the Phillips
curve from 1958 onwards by evaluating the models proposed in Amisano and Giacomini (2007).
We conclude in section 5. Appendix A contains mathematical proofs and Appendix B a description
of the parametric bootstrap to correct the size of the tests. We also provide a supplementary file
with additional simulation material.
2 Statistics and Asymptotic Distributions
2.1 Construction of the Statistics
The test statistics are based on the autocontour (ACR) and generalized autocontour (G-ACR)
methodology proposed by Gonzalez-Rivera et al. (2011, 2012, 2014) that provides powerful tests
for dynamic specification of the conditional density model either in-sample or out-of-sample en-
vironments. In the present context, we adapt these tests to instances where instabilities may be
present in the data so that, beyond the evaluation of the density model, we will also be able to
detect unstable periods.
Let Yt denote the random process of interest with conditional density function f(yt|Ωt−1), where
Ωt−1 is the information set available up to time t− 1. Observe that the random process Yt could
enjoy very general statistical properties, e.g. heterogeneity, dependence, etc. The researcher will
construct the conditional model by specifying a conditional mean, conditional variance or other
conditional moments of interest, and making distributional assumption on the functional form of
3
f(.). Based on the conditional model, she will proceed to construct a density forecast. If the
proposed predictive density model for Yt, i.e. f ∗t (yt|Ωt−1)Tt=1 coincides with the true conditional
density ft(yt|Ωt−1)Tt=1, then the sequence of probability integral transforms (PIT) of YtTt=1 w.r.t
f ∗t (yt|Ωt−1)Tt=1 i.e. utTt=1 must be i.i.d U(0, 1) where ut =∫ yt−∞ f
∗t (vt|Ωt−1)dvt. Thus, the null
hypothesis H0 : f ∗t (yt|Ωt−1) = ft(yt|Ωt−1) is equivalent to the null hypothesis H′0 : utTt=1 is i.i.d
U(0, 1) (see Rosenblatt (1952), Diebold et al., 1998).
In order to provide a self-contained exposition, we briefly review the approach in Gonzalez-Rivera
and Sun (2014). We start by constructing the generalized autocontours (G-ACR) under i.i.d.
uniform PITs. Under H′0 : utTt=1 i.i.d U(0, 1), the G-ACRαi,k is defined as the set B(.) of points
in the plane (ut, ut−k) such that the square with√αi-side contains at most αi% of observations,
i.e.,
G-ACRαi,k = B(ut, ut−k) ⊂ <2‖ 0 ≤ ut ≤√αi and 0 ≤ ut−k ≤
√αi, s.t. : ut × ut−k ≤ αi
We construct an indicator series Ik,αit as follows
Ik,αit = 1((ut, ut−k) ⊂ G-ACRαi,k) = 1(0 ≤ ut ≤√αi, 0 ≤ ut−k ≤
√αi)
Based on this indicator, Gonzalez-Rivera and Sun (2014) proposed the following t-tests and chi-
square statistics to test the null hypothesis H′0 : utTt=1 i.i.d U(0, 1), which is equivalent to testing
for correct specification of the full conditional density model.
(1) t-ratio testing: for fixed autocontour αi and fixed lag k,
z ≡√T − k(αi − αi)
σk,i
d−→ N(0, 1)
where αi =∑Tt=k+1 I
k,αit
T−k , and σ2k,i is the asymptotic variance of αi.
4
(2) chi-squared testing:
(2.1) For a fixed lag k, C′kΩ−1k Ck
d−→ χ2C where Ck = (ck,1, ...ck,C)
′is a C × 1 stacked vector with
element ck,i =√T − k(αi − αi), and Ωk the asymptotic variance-covariance matrix of the vector
Ck.
(2.2) For a fixed autocontour αi, L′αiΛ−1αi
Lαid−→ χ2
K where Lαi = (`1,αi , ...`K,αi)′
is a K×1 stacked
vector with element `k,αi =√T − k(αi−αi), and Λαi is the asymptotic variance-covariance matrix
of the vector Lαi .
In a potentially unstable environment, we will construct the tests within the following rolling
sample scheme. The total sample size T is divided into two parts: in-sample observations (R)
and out-of-sample observations (P). We form subsamples of size r from t − r + 1 up to t, where
t = R + r, · · · , T . In each subsample, we evaluate the proposed predictive density by calculating
the three statistics reviewed in (1), (2.1), and (2.2), which we call z, C and L. As a result, we
obtain three sets of n ≡ T − r − R + 1 tests each i.e., zjnj=1, Cjnj=1 and Ljnj=1. Finally, to
detect instabilities, we construct Sup-type and Avg-type statistics by taking the supremum (S)
and the average (A) respectively over each set |zj|nj=1, Cjnj=1 andLjnj=1 so that we obtain the
following six statistics: S|z|, SC , SL and A|z|, AC , AL.
2.2 Asymptotic Properties of the Statistics
Under the following set of assumptions, we provide three propositions, which proofs are provided
in the Appendix A.
• A1: For T → ∞, R → ∞, P → ∞, limT→∞PR
= 0 ; and for r, P → ∞, limT→∞r−kP
= m,
where r is the size of the rolling subsample in the evaluation set, m ∈ (0, 1), and k is the lag
in the Ik,αit indicator. 2
2According to several Monte Carlo studies (West, 2006, and the references herein), the ratio P/R is consideredsufficiently small and parameter uncertainty will be almost negligible when P/R ≤ 10%. When limT→∞
PR 6= 0,
5
• A2: E|Ik,αit |q < ∆ <∞ for some q ≥ 2. This assumption is trivial as the second moments of
the indicator are well defined as we will see next.
• A3: The data yt comes from a stationary and ergodic process. 3
Proposition 1 Let J be the index that identifies a particular subsample in the evaluation period,
i.e. J = [Ps], s ∈ [m, 1], [Pm] = r − k, and let αi(J) =∑R+Jt=R+1+J−r+k I
k,αit
r−k be the corresponding
subsample proportion based on the indicator. The Sup- and Avg-tests are
S|z| = supJ
∣∣∣∣√r − k(αi(J)− αi)σk,αi
∣∣∣∣A|z| =
1
P − r + 1
J∑J
∣∣∣∣√r − k(αi(J)− αi)σk,αi
∣∣∣∣where σ2
k,αi= αi(1− αi) + 2α
3/2i (1− α1/2
i )
Given assumptions A1-A3, and under the null hypothesis of i.i.d U(0, 1) PITs, the asymptotic
distribution of the tests, for P →∞, is the following
S|z|d−→ sup
s∈[m,1]
|W (s)−W (s−m)|√m
A|z|d−→
∫ s
s
|W (s)−W (s−m)|√m
ds
where W (.) is a standard univariate Brownian motion.
parameter uncertainty will affect the asymptotic variance σ2k,αi
of the statistic. By taking a mean value expansionaround the true parameter values θ0 and applying the Slutsky’s theorem we obtain
√P (αi(θ)− αi) =
√P (αi(θ0)− αi) +
√P√R
√R(θ − θ0)′ lim
R→∞E∂αi
∂θ|θ=θ0+ op(1)
The variance of the second term of the expansion as well as the covariance between the first and second terms willcontribute to the asymptotic variance of the statistics. For calculation of these terms, see Gonzalez-Rivera, Senyuz,and Yoldas (2011). In practice, these terms will be calculated by a fully parametric bootstrap procedure.
3This assumption can be relaxed to include more general mixing processes because the relevant conditions toinvoke limiting theorems as the FCLT are those of the indicator process.
6
Proposition 2 For a fixed lag k, write ck,i(J) =√r − k(αi(J)−αi) and stack ck,i(J) for different
autocontours levels i = 1, 2, ...C such that Ck(J) = (ck,1(J), ...ck,C(J))′is the C×1 stacked vector.
The Sup- and Avg-tests are
SC = supJ
Ck(J)′Ω−1k Ck(J)
AC =1
P − r + 1
J∑J
Ck(J)′Ω−1k Ck(J)
where Ωk is the asymptotic variance and covariance matrix of the random vector Ck(J). Given
assumptions A1-A3, and under the null hypothesis of i.i.d U(0, 1) PITs, the asymptotic distribution
of the tests, for P →∞, is the following
SCd−→ sup
s∈[m,1]
(W(s)−W(s−m))′(W(s)−W(s−m))
m
ACd−→
∫ s
s
(W(s)−W(s−m))′(W(s)−W(s−m))
mds
where W(.) is a standard C-variate Brownian motion.
Proposition 3 For a fixed autocontour αi, write `k,αi(J) =√r − k(αi(J) − αi) and stack `k,αi
for k = 1, ....K. Let Lαi(J) = (`1,αi(J), ...`K,αi(J))′
be the K × 1 stacked vector. The Sup- and
Avg-tests are
SL = supJ
Lαi(J)′Λ−1αi
Lαi(J)
AL =1
P − r + 1
J∑J
|Lαi(J)′Λ−1αi
Lαi(J)|
where Λαi is the asymptotic variance and covariance matrix for the random vector Lαi . Given
assumptions A1-A3, and under the null hypothesis of i.i.d U(0, 1) PITs, the asymptotic distribution
7
of the tests, for P →∞, is the following
SLd−→ sup
s∈[m,1]
(W(s)−W(s−m))′(W(s)−W(s−m))
m
ALd−→
∫ s
s
(W(s)−W(s−m))′(W(s)−W(s−m))
mds
where W(.) is a standard K-variate Brownian motion.
We tabulate the percentiles of the asymptotic distributions of the tests provided in Propositions 1 to
3. Since the distributions depend onm, which is proportion of the rolling sample to the total evalua-
tion sample, we consider the following values of m ∈ [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9]. In Table
1, we report the percentiles of the distributions of the S|z| and A|z| statistics; in Table 2, those for the
SC andAC considering the 13 autocontour C = [0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99];
and in Table 3 those for the SL and AL tests considering 5 lags.
[TABLES 1-3 ABOUT HERE]
Since a model can be considered misspecified whenever does not take into account breaks in the
data, the proposed stability tests can be also viewed as statistics assessing jointly the specification
and the time stability of the model. In practice, the tests will be most useful when models that
are estimated with R observations, need to be validated over a new set of observations, in our
case, over the prediction sample of P observations, and in doing so, we assess whether the model
is stable over different time periods. In the following sections, Monte Carlo simulations and the
evaluation of the Phillips curve, we show how to use the stability tests.
3 Monte Carlo Simulations
We perform extensive Monte Carlo simulations to assess the finite sample properties (size and
power) of the proposed statistics. For√R-consistent estimators of the parameters of the model,
8
i.e. (θ − θ0) = Op(R−1/2) with a well-defined asymptotic distribution, and under assumption A1,
parameter uncertainty is asymptotically negligible. In this case, the critical values tabulated in
Tables 1 to 3 can be used directly. Whenever the condition P/R → 0 is violated, we need to
adjust the asymptotic variance of the tests, otherwise the size of the tests will be distorted. As
we mention in the previous section, the adjustments to the variance will be difficult to calculate
analytically because they are model dependent. For practical purposes, we propose to implement
a fully parametric bootstrap procedure because the density model is fully known under the null
hypothesis of correct dynamic specification and correct functional form of the conditional density,
i.e., stability of the density model over the prediction sample. The bootstrap procedure entails
either to bootstrap the asymptotic variance and use the tabulated critical values or bootstrap
directly the distribution of the proposed Sup- and Ave- tests to obtain their critical values. In
the Appendix B, we explain how we implement the parametric bootstrap to obtain bootstrapped
critical values. Nevertheless, in the following simulations, we also explore the effect of different
values of the ratio P/R on the size of the tests. We consider fixed, rolling, and recursive forecasting
schemes.
3.1 Size of the Tests
Under the null hypothesis of a stable density model, we consider the following data generating
process: yt = α1 + β1yt−1 + β2xt−1 + σεt where xt = φ1 + φ2xt−1 + νt, and εt and νt ∼ N(0, 1),
φ1 = 1.38, φ2 = 0.77, α1 = 1.5, β1 = 0.5, β2 = 0.6, σ = 1. We consider sample sizes of T = 150
(with evaluation sample P=60), T = 375 (with P=150), and T = 750 (with P=300) observations,
and for each sample size, we consider the proportion m = r/P equal to m = 1/3, m = 1/2 and
m = 2/3. We maintain the ratio P/R constant and equal to 2/3. In total, we run nine experiments,
of which we present here the two most extremes: small sample size with small subsample window
(T = 150,m = 1/3) and large sample size with large subsample window (T = 750,m = 2/3). The
9
size results of these two cases are presented in Tables 4 and 5 respectively. The remaining seven
cases are available in a “supplementary material” file. We work with 13 autocontour coverage
levels C = [0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99]. The number of Monte Carlo
replications is 1,000, the number of bootstrap samples is 500, and the nominal size is 5%.
[TABLES 4-5 ABOUT HERE]
The overall size of the tests is very good in the nine experiments considered. There are not
substantial differences among the fixed, rolling, and recursive estimation schemes. We find some
small size distortions (under-sized) when the sample is small (T = 150) and the autocontour levels
are extreme (1% and 99%) but as the sample size increases, the distortion disappears. For the
individual tests Sk,αi|z| and Ak,αi|z| (k and αi fixed), the Ave-test tends to have better size than the
Sup-test. The tests SC , AC , SL, and AL have very good size even in small samples.
For the same generating process explained in the above paragraphs, with a small estimation sample
R = 96 and a small subsample window m = 1/3, we have considered three increasing values for
the ratio P/R, i.e., 1/4 (T = 120), 1/2 (T = 144), and 3/4 (T = 168). The size results for
these three cases are presented in Tables 6 to 8. For the case of P/R = 1/4, we observe that the
individual tests Sk,αi|z| are more undersized than the tests Ak,αi|z| , for which the average size is about
4%. The portmanteau tests SC , AC , SL, and AL have a size between 3 and 5%. Note that for this
case we have a very small prediction sample P = 24 and the subsample window includes only 8
observations. For the case of P/R = 1/2, there is an overall improvement of the size for all tests,
and for the case P/R = 3/4, most tests have a size around the nominal size of 5%. It seems that
the small distortions that we observe are due to the small number of observations in the subsample
window. As we increase the ratio P/R, the parametric bootstrap is properly taking care of the
effect of parameter uncertainty in the variance of the tests and delivering tests with the correct
size.
[TABLES 6-8 ABOUT HERE]
10
3.2 Power of the Tests
To assess the power of the tests, we generate data from four different processes, all of them
containing a break point. The model that we maintain under the null hypothesis is the same
as the one considered in the study of the size properties: yt = α1 + β1yt−1 + β2xt−1 + σεt with
xt = φ1 +φ2xt−1 +νt, νt and εt ∼ N(0, 1). The total sample size (T ) is 650, R = 350, P = 300, and
m = 1/3. The break point happens at at R + τP , where τ = 1/3, 1/2, and 2/3. In the following
experiments, the number of Monte Carlo replications is 1000 and bootstrapped samples 500. We
maintain a nominal test size of 5%.
The four data generating mechanisms are the following:
DGP1: Break in the intercept of yt = αt + β1yt−1 + β2xt−1 + σεt, εt ∼ N(0, 1):
αt =
α1 = 1.5 if t < break
α2 = 2 otherwise
with β1 = 0.5, β2 = 0.6, σ = 1.
DGP2: Break in the variance of yt = α + β1yt−1 + β2xt−1 + σtεt, εt ∼ N(0, 1):
σt =
σ1 = 1.5 if t < break
σ2 = 1.8 otherwise
with α = 1.5, β1 = 0.5, β2 = 0.6.
DGP3: Breaks in the slope coefficients of yt = α + β1,tyt−1 + β2,txt−1 + σεt, εt ∼ N(0, 1):
β1,t =
β1,1 = 0.5 if t < break
β1,2 = 0.3 otherwise
11
β2,t =
β2,1 = 0.6 if t < break
β2,2 = 0.4 otherwise
with α = 1.5, σ = 1.
DGP4: Breaks in the intercept, variance and slope coefficients of yt = αt+β1,tyt−1 +β2,txt−1 +σtεt,
εt ∼ N(0, 1):
αt =
α1 = 1.5 if t < break
α2 = 2 otherwise
σt =
σ1 = 1.5 if t < break
σ2 = 1.8 otherwise
β1,t =
β1,1 = 0.5 if t < break
β1,2 = 0.3 otherwise
β2,t =
β2,1 = 0.6 if t < break
β2,2 = 0.4 otherwise
Note that the breaks considered are not very extreme. We perform all the simulations under fixed,
rolling, and recursive estimation schemes. We report the power results for the fixed and rolling
schemes for the four DGPs with τ = 1/3 in Tables 9 to 12 and those for the recursive scheme in
the supplementary material.
[TABLES 9-12 ABOUT HERE]
Under the fixed scheme, the tests are most powerful (power of about 90%) to detect breaks in
intercept and slope coefficients (DGP1, DGP3, and DGP4). Both Ave- and Sup- tests enjoy
similar performance, either for single hypothesis (S|z| and A|z|) or for joint hypothesis (SC , AC ,
SL, AL). The power drops when testing for breaks in the variance (DGP2) with values of 10-60%
for the single hypothesis tests and of 60% for the joint hypothesis tests. Under the rolling scheme,
12
as we expected the tests are overall less powerful because by rolling the estimation sample, the
model adjusts slowly to the new parameters values. Nevertheless, the power of SC , AC , SL and
AL is still very high (around 50-80%) for DGP1, DGP3, and DGP4, and around 40% for DGP2.
For all DGPs, the Ave-test is more powerful that the Sup. Under the recursive scheme, the tests
performance is slightly worse than in the fixed scheme but slightly better than in the rolling scheme.
We move the location of the break, i.e., τ = 1/2 and 2/3 and we present the power results for
DGP3 with a rolling scheme in Table 13. The overall finding is that the power of the tests decreases
as the break point moves towards the end of the prediction sample, which is expected. However,
the decrease in power is not very abrupt. For instance, for the AC test the power moves from
about 80% (τ = 1/3) to 65% (τ = 1/2) to 55% (τ = 2/3)
[TABLE 13 ABOUT HERE]
3.3 Comparison with Rossi & Sekhposyan tests
We compare our tests with those of Rossi & Sekhposyan (R&S) (2013). In both instances, we
entertain the same null hypothesis, i.e. the constancy of the density model (specification of the
model and the functional form of the conditional density) over the prediction sample. R&S tests
follow the set up of Corradi and Swanson (2006) so that their tests, also based on the PITs
of the proposed model, are a function of the distance between the empirical cumulative distri-
bution function and that of the uniform distribution, which is a 45 degree line. The κP test is a
Kolmogorov-Smirnov-type statistics and the CP is a Cramer-von Mises-type statistic. Our G-ACR
tests are based on the object ”autocontour” and measure the dense-ness and uniform distribution
of the PITs over a collection of squares in a two-dimensional space (ut−k, ut). By construction,
our tests exploit the independence properties of the PITs directly. The asymptotic distribution of
the R&S tests is based on the statistical properties of an empirical process. Our tests are simpler
in that rely on the properties of a binary indicator with well-defined moments. When parameter
13
uncertainty is non-negligible, the critical values of the R&S tests and the G-ACR tests are obtained
by simulation.
We follow the same experimental design explained in section 4.2 of R&S article. Their tests
κP and CP are directly comparable with our tests SC and AC stacking all the 13 autocontour
levels. We consider the same DGPs as in Table 2 of R&S article: DGP1 considers the case of
misspecification in the functional form of the density, DGP2 considers instability in the mean and
variance parameters, and DGP3 considers time-varying distortions in the shape of the assumed
conditional density. We run the experiments with R = 500, P = 100 and a rolling ratio m = 0.8
in the prediction sample. The models are estimated recursively. The number of Monte Carlo
replications is 1000 with 500 bootstrap samples.
When c = 0 (DGP1 and DGP2) and T1/T = 600 (DGP3), the respective DGPs coincide with the
model under the null hypothesis, so the rejection frequency is equal to the size of the tests. The
larger the value of c of the lowest the ratio T1/T , the furthest the DGP model is from the model
under the null. All tests have good sizes but the S1,13C and A1,13
C tests are more powerful across
DGPs, in particular for the cases of DGP2 (c = 0.5 and c = 1) and DGP3 (T1/T = 580 and 560).
14
Empirical Rejection Frequencies
DGP1
c κP CP S1,13C A1,13
C
0 0.04 0.04 0.05 0.040.5 0.04 0.05 0.07 0.071 0.13 0.14 0.21 0.20
1.5 0.30 0.34 0.48 0.402 0.50 0.60 0.71 0.63
2.5 0.73 0.82 0.87 0.84
DGP2
c κP CP S1,13C A1,13
C
0 0.04 0.04 0.05 0.050.5 0.16 0.14 0.53 0.501 0.66 0.47 0.99 0.97
1.5 0.97 0.92 1.00 1.002 1.00 1.00 1.00 1.00
2.5 1.00 1.00 1.00 1.00
DGP3
T1/T κP CP S1,13C A1,13
C
600 0.04 0.04 0.05 0.05580 0.27 0.29 0.73 0.50560 0.78 0.78 1.00 0.99540 0.96 0.97 1.00 1.00
4 Density Forecast Evaluation of the Phillips Curve
We apply the proposed tests to analyze the stability of the Phillips curve. Stock and Watson
(1999) found some empirical evidence in favor of the Phillips curve as a forecasting tool, they
showed that inflation forecasts produced by the Phillips curve were more accurate than forecasts
based on simple autoregressive or multivariate models but they also found parameter instabilities
across different subsamples. Rossi and Skehposyan (2010) showed that the predictive power of
the Phillips curve disappeared around the time of the Great Moderation. Based on scoring rules,
Amisano annd Giacomini (2007) compared the density forecast accuracy of several models of the
15
Phillips curve and concluded that the best density forecast is produced by a Markov-switching
model. Since their comparisons are based on the average forecasting performance of competing
models over time, they cannot directly address the instabilities widely documented in the literature.
In this section, we consider the models in Amisano annd Giacomini (2007) and we focus on their
absolute density forecast performance in the presence of instabilities. Our starting model is a
linear Phillips curve (Stock and Watson, 1999), in which changes of the inflation rate depend on
their lags and on lags of the unemployment rate i.e.,
∆πt = α1 + β1∆πt−1 + β2∆πt−2 + β12∆πt−12 + γut−1 + σεt
where πt = 100 × ln(CPIt/CPIt−12); CPIt is the consumer price index for all urban consumers
and all items; ut−1 is the civilian unemployment rate; and εt ∼ N(0, 1). The data is collected from
the FRED database; monthly CPI and unemployment series are both seasonally adjusted. The
time series run from 1958M01 to 2012M01 (updated sample from 1958M01-2004M07 in Amisano
and Giacomini). Standard tests on ∆πt and ut show that they do not have a unit root. On
implementing our tests, we consider the same estimation sample as in Amisano and Giacomini,
from 1958M01 to 1987M12 (360 observations). The evaluation sample runs from 1988M01 to
2012M01 (289 observations) with subsamples of size r = 200.
4.1 Evaluation of the Linear Phillips Curve
We present the evaluation results for the linear Phillips Curve in Table 14 under fixed and rolling
estimation schemes. The recursive case is in the supplementary material.
[TABLE 14 ABOUT HERE]
The portmanteau tests SC , AC , SL, and AL indicate a clear rejection of the linear model. On
examining the individual tests S|z| and A|z|, the rejection comes from the middle autocontours,
16
between 40% and 70% coverage, and from the 95-99% levels (large changes in inflation). In Figure
1, we plot the t- and C-statistics over the evaluation period in a sequential fashion. The tests
break through their corresponding critical values around the 60th statistic, and the values of the
tests keep on increasing reaching two local maxima, which points to two potential breaks: the first
in the 64th statistic for the t-tests (1993M03 to 2009M11) and in the 71th statistic for the C tests
(1993M11 to 2010M06), and the second local maximum in the 78th statistic that corresponds to
the period 1994M06 to 2011M01. All these periods include the years 1993-2007 in the so-called
Great Moderation and the years after the deep financial crisis of 2008.
0 10 20 30 40 50 60 70 80 900
1
2
3
4
5
6
7
0 10 20 30 40 50 60 70 80 9010
20
30
40
50
60
t statisticscritical values
Chi statisticscritical values
Figure 1: Plots of t-ratio (99% autocontour) and C Statistics for Linear Phillips Curve(fixed scheme)
4.2 Evaluation of Non-Linear Phillips Curve
Given the rejection of the linear Phillips curve, we proceed with a flexible specification by assuming
that the coefficients in the linear model vary according to a Markov switching mechanism. We
17
consider a two-state Markov switching model, i.e.,
∆πt = αst + βst1 ∆πt−1 + βst2 ∆πt−2 + βst12∆πt−12 + γstut−1 + σstεt
where the unobserved state variable st switches between two states,1 or 2, with transition proba-
bilities Pr(st = j|st−1 = i) = pij for i, j = 1, 2; and εt is assumed to be a standard normal variate.
Thus, this model allows for non-Gaussian density forecasts. Since all the coefficients depend on the
state variable (Model 1), the model is extremely flexible and it will adapt to potential breaks or
instabilities that may occur over time. We have run our test statistics and we fail to reject the null
hypothesis of correct specification. However, we would like to investigate what coefficients are key
to understand where the nonlinear behavior comes from. We consider six additional specifications,
Model 2: ∆πt = α+βst1 ∆πt−1 +βst2 ∆πt−2 +βst12∆πt−12 +γut−1 +σεt, (intercept and unemployment
coefficient do not depend on st).
Model 3: ∆πt = αst + β1∆πt−1 + β2∆πt−2 + β12∆πt−12 + γstut−1 + σstεt, (inflation coefficients do
not depend on st).
Model 4: ∆πt = α+ β1∆πt−1 + β2∆πt−2 + β12∆πt−12 + γut−1 + σstεt, (only the standard deviation
depends on st).
Model 5: ∆πt = αst + β1∆πt−1 + β2∆πt−2 + β12∆πt−12 + γstut−1 + σεt, (only the intercept and
unemployment coefficient depend on st).
Model 6:∆πt = αst + β1∆πt−1 + β2∆πt−2 + β12∆πt−12 + γut−1 + σstεt, (only the intercept and the
standard deviation depend on st).
Model 7:∆πt = α + β1∆πt−1 + β2∆πt−2 + β12∆πt−12 + γstut−1 + σεt, (only the unemployment
coefficient depends on st).
Out of these six specifications, we observe outright rejections at the 5% significance level for
18
Model 2 (rejection in the 90-99% autocontours), Model 4 (rejection in the 95-99% autocontours),
and Model 7 (rejection in the 30-50% autocontours). We only fail to reject Model 3 in which we
allow the intercept,the unemployment coefficient, and the standard deviation of the error to be
state-dependent while the rest of the parameters are constant. In this case, all statistics (joint
and individual) enjoy p-values larger than 5%. Models 5 and 6 are close contenders to Model 3.
In Model 5, where only the intercept and the unemployment coefficient are state-dependent, we
observe a marginal rejection (p-values around 4%) with the S|z| and A|z| tests for the 1 and 5%
autocontours. In Model 6, where only the intercept and the standard deviation of the error are
state-dependent, we observe a marginal rejection (p-values around 4%) with the S|z| test for the
1% autocontour. The overall message of these testing results is that a shift in the level αst and
a shift in the standard deviation σst (Model 6) may be enough to understand the instability of
the Phillips curve but if, in addition, we let the coefficient that links inflation and unemployment
to be state-dependent (Model 3), the density forecast of inflation changes will be more robust to
unstable times. We report the testing results for Models 3 and 6 in Table 15.
[TABLE 15 ABOUT HERE]
For the preferred model, Model 3, we compute the natural rate of unemployment (NAIRU), which
is the ratio between (minus) the intercept term and the coefficient on unemployment. In recession
times (state 1), the estimated NAIRU is 7.52% and in expansions (state 2), 5.53%. The persistence
of state 1 is p11 = 0.96 and that of state 2 is p22 = 0.99 so that recessions are shorter than
expansions. The estimated variance of state 1 is 0.20 and that of state 2 is 0.04 so that recessions
are more volatile than expansions.
19
5 Conclusion
We have provided a battery of tests to assess the stability of the density forecast over time, which
offer important advantages for the empirical researcher. These tests are nuisance-parameter free
and their asymptotic distributions can be tabulated. If the tests reject the null hypothesis of a
stable density forecast, the shapes of the empirical generalized autocontours can be visualized to
extract information regarding the direction of the rejection. Regardless of the estimation scheme
(fixed, rolling, or recursive), their finite sample properties are superior. In some instances, the
Ave-tests tend to have a slightly better size than the Sup-tests. Both types are more powerful on
detecting breaks in the intercept and slope coefficients than on detecting breaks in the variance.
The tests can also be easily applied to multivariate random processes of any dimension. We have
compared our tests with those of Rossi and Sekhposyan (2013) and found that our G-ACR tests
are more powerful across the DGPs considered. As an application of the tests, we have analyzed
the stability of the Phillips curve. A linear model is strongly rejected in favor of a non-linear
specification that allows shifts in the mean and in the variance of the inflation changes as well as
in the coefficient linking inflation and unemployment. The break of the linear model occurs during
the Great Moderation years.
20
Appendix A
Proof of Proposition 1:
The indicator function Ik,αit is a Bernoulli random variable with the following moments: E(Ik,αit ) =
αi, V ar(Ik,αit ) = αi(1− αi) and covariance
rαih ≡ cov(Ik,αit , Ik,αit−h ) =
0 if h 6= k
α3/2i (1− α1/2
i ) if h = k
Since the indicator process is stationary and ergodic, αi(J) satisfies the condition of global covari-
ance stationarity required for the FLCT to apply (Theorem 7.17) in White (2001). Since J = [Ps],
s ∈ [m, 1], and r − k = [Pm], we write as P →∞
WP (s) ≡√
(r − k)(αi(J)− αi)σk,i
=
∑R+Jt=R+1+J−r+k(I
k,αit − αi)√
r − kσk,i
=
∑R+Jt=R+1(Ik,αit − αi)√
(r − k)σk,i−∑R+J−(r−k)
t=R+1 (Ik,αit − αi)√(r − k)σk,i
=
√P√Pm
∑R+[Ps]t=R+1 (Ik,αit − αi)√
Pσk,i−√P√Pm
∑R+[P (s−m)]t=R+1 (Ik,αit − αi)√
Pσk,id−→ 1√
m(W (s)−W (s−m))
where W (.) is the standard Brownian motion and the limiting distribution, in the last line, is a
direct consequence of the FCLT. Finally, by the Continuous Mapping Theorem, we have:
S|t| = supJ|√r − kαi(J)
σk,αi| d−→ sup
|W (s)−W (s−m)|√m
A|t| =1
P − r + 1
J∑J
|√r − kαi(J)
σk,αi| d−→
∫ s
s
|W (s)−W (s−m)|√m
21
where J = [Ps] and J = [P s].
Proof of Proposition 2
Let Ωk be the variance-covariance matrix of Ck(J) whose typical element ωi,j is calculated as
follows
cov(ck,i, ck,j) = cov(Ik,αit , Ik,αjt ) + cov(Ik,αit , I
k,αjt−k ) + cov(Ik,αit−k , I
k,αjt ) + o(1)
If i = j, by Proposition 1, ωi,i = var(√T − k(αi − αi)) = αi(1 − αi) + 2α
3/2i (1 − α1/2
i ). If i < j,
αi < αj, and we have
cov(Ik,αit , Ik,αjt ) = E(Ik,αit × Ik,αjt )− αi × αj = αi(1− αj)
cov(Ik,αit , Ik,αjt−k ) = E(Ik,αit × Ik,αjt−k )− αi × αj = αi × α1/2
j − αi × αj
cov(Ik,αit−k , Ik,αjt ) = E(Ik,αit−k × I
k,αjt )− αi × αj = αi × α1/2
j − αi × αj
If i > j, the above expressions hold by just switching the subindexes i and j.
Since the vector Ck(J) is globally stationary, we can invoke a multivariate FLCT (see Theorem
7.29 and 7.30 in White (2001)). By following the same arguments as in Proposition 1, we have as
P →∞,
WP (s) ≡ Ω−1/2k Ck(J)
d−→ 1√m
(W(s)−W(s−m))
where W(s) is a C-variate Brownian process. By the Continuous Mapping Theorem, we have
SC = supJ
Ck(J)′Ω−1k Ck(J)
d−→ sups∈[m,1]
(W(s)−W(s−m))′(W(s)−W(s−m))
m
22
and
AC =1
P − r + 1
J∑J
|Ck(J)′Ω−1k Ck(J)|
d−→∫ s
s
(W(r)−W(r −m))′(W(r)−W(r −m))
m
Proof of Proposition 3: Let Λαi be the variance-covariance matrix of Lαi(J) whose typical
element λj,k is calculated as
λj,k =
αi(1− αi) + 2α3/2i (1− α1/2
i ) if j = k
4α3/2i (1− α1/2
i ) if j 6= k
Therefore, the vector Lαi(J) is globally stationary, and we can invoke a multivariate FLCT (see
Theorems 7.29 and 7.30 in White (2001)). By following the same arguments as in Proposition 1,
we have as P →∞,
WP (s) ≡ Λ−1/2αi
Lαi(J)
d−→ 1√m
(W(s)−W(s−m))
where W(s) is a L-variate Brownian process. By the Continuous Mapping Theorem, we have
SL = supJ
Lαi(J)′Λ−1
αiLαi
(J)
d−→ sups∈[m,1]
(W(s)−W(s−m))′(W(s)−W(s−m))
m
23
AL =1
P − r + 1
J∑J
Lαi(J)′Λ−1
αiLαi
(J)
d−→∫ s
s
(W(s)−W(s−m))′(W(s)−W(s−m))
m
24
Appendix B
Description of the Parametric Bootstrap
Suppose that we are interesting in calculating the size of the test. Consider a DGP, e.g, yt =
α + β1yt−1 + β2xt−1 + σεt with population parameters α = 1.5, β1 = 0.5, β2 = 0.6, σ = 1 and
εt → N(0, 1). The information on xt is predetermined, so it is known in advance. Assume a sample
size, e.g., T = 750, that is split as R = 450 (estimation sample) and P = 300 (prediction sample),
and r = 100 (m = 1/3) is the rolling sample over the prediction sample. We run 1,000 Monte
Carlo replications, and 500 bootstrap samples within each Monte Carlo replication.The Monte
Carlo/bootstrap procedure consists of the following step:
1. One Monte Carlo replication: draw from the conditional density of εt, e.g. from N(0, 1). With
the population parameters inserted into the DGP, generate one Monte Carlo sample of observations
yt.
2. With the first replication of observations yt, estimate the model parameters, e.g., θ =
(α, β1, β2, σ) . Obtain the sequence of one-step-ahead density forecasts, i.e. ft(yt|Ωt−1, θ) and
the corresponding PITs, i.e. ut =∫ yt−∞ ft(νt|Ωt−1, θ)dνt for the first subsample of size 100 in the
prediction sample. With these PITs, construct the tests |z1|, C1, and L1. By rolling one observa-
tion at the time over the prediction sample, we have now a second subsample of size 100 and we
construct again the one-step-ahead density forecasts, the corresponding PITs, and tests |z2|, C2,
and L2. If we keep rolling the observations, for r = 100 over P = 300, we will have 201 statistics,
that is, |zj|201j=1, Cj201
j=1, and Lj201j=1. Over these collections of tests, we calculate the supremum
(S) and the average (A) to obtain the following six statistics S|z|, SC , SL and A|z|, AC , AL.
In order to take care of parameter uncertainty, we implement a bootstrap procedure within each
of the Monte Carlo replications. We draw 500 bootstrap samples as follows:
3. We fixed the parameters to the estimated values in step 2, e.g., θ = (α, β1, β2, σ) and draw from
25
the conditional density of εt, e.g. from N(0, 1) to generate bootstrap samples of ybt. For each
bootstrap sample, we proceed to estimate the parameters again θb = (αb, β1b, β2
b, σb) and, as in
step 2, we construct density forecasts, PITs, and the collection of tests |zj|b201j=1, Cb
j201j=1, and
Lbj201j=1 to finally obtain the statistics Sb|z|, S
bC , SbL and Ab|z|, A
bC , AbL. We repeat 500 times and we
obtain the empirical distribution (quantiles) of each of the six statistics. For instance, we could
compute the bootstrap critical value or 95% percentile for each test over the 500 samples. Suppose
that we call the 95% percentile Sb(95%)|z| for the tests Sb|z|500
b=1, and similarly for the other statistics.
4. Given the empirical distribution (over the 500 bootstrap samples) of each statistic, we bring
the values of the tests computed in step 2, i.e. S|z|, SC , SL and A|z|, AC , AL, and compare them
with the bootstrapped critical values obtained in step 3. For instance, if Sb(95%)|z| ≤ S|z|, we count
it as one, otherwise as a zero.
Observe that steps 2, 3, and 4 are carried out for one Monte Carlo replication. Thus,
5. Repeat steps 1, 2, 3, and 4 for 1000 Monte Carlo replications and record the number of ones
(step 4) across the 1000 replications. The size of the test will be the percentage number of ones
over 1000.
26
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28
Tables: Asymptotic Distributions
In the following six tables, the percentiles are obtained from 2000 replications with a sample sizeof 20,000 observations in each replication.
Asymptotic Distribution of S|z| Statisticm
Percentile 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.999% 3.950 3.713 3.510 3.396 3.428 3.213 3.089 3.112 2.95195% 3.502 3.267 3.066 2.926 2.866 2.679 2.537 2.439 2.40690% 3.292 2.987 2.845 2.642 2.571 2.361 2.240 2.121 2.01580% 3.020 2.684 2.522 2.372 2.217 2.053 1.935 1.849 1.65570% 2.843 2.489 2.315 2.156 1.999 1.834 1.697 1.582 1.39260% 2.702 2.343 2.145 1.983 1.819 1.653 1.529 1.383 1.19750% 2.569 2.203 2.015 1.830 1.664 1.515 1.365 1.221 1.02140% 2.457 2.090 1.884 1.684 1.515 1.373 1.212 1.066 0.86730% 2.339 1.975 1.747 1.540 1.379 1.228 1.071 0.915 0.73020% 2.201 1.822 1.586 1.384 1.226 1.074 0.923 0.791 0.60610% 2.033 1.629 1.420 1.199 1.055 0.901 0.761 0.631 0.4795% 1.922 1.492 1.290 1.065 0.943 0.813 0.663 0.535 0.3941% 1.730 1.265 1.072 0.829 0.739 0.631 0.498 0.409 0.287
Asymptotic Distribution of A|z| Statisticm
Percentile 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.999% 1.241 1.435 1.656 1.824 2.204 2.181 2.360 2.542 2.58395% 1.078 1.199 1.355 1.487 1.697 1.694 1.774 1.854 1.99890% 1.004 1.088 1.206 1.300 1.418 1.406 1.490 1.586 1.66680% 0.918 0.970 1.035 1.101 1.115 1.128 1.183 1.264 1.25670% 0.870 0.891 0.922 0.948 0.939 0.928 0.966 1.017 1.01960% 0.825 0.825 0.827 0.837 0.818 0.796 0.807 0.824 0.82550% 0.789 0.760 0.760 0.741 0.710 0.685 0.682 0.675 0.66140% 0.752 0.706 0.690 0.653 0.615 0.581 0.551 0.534 0.50730% 0.711 0.649 0.617 0.568 0.524 0.487 0.453 0.417 0.37820% 0.673 0.589 0.544 0.497 0.451 0.409 0.358 0.320 0.27410% 0.608 0.517 0.465 0.409 0.359 0.319 0.269 0.234 0.1865% 0.567 0.458 0.413 0.342 0.310 0.260 0.219 0.188 0.1401% 0.478 0.381 0.325 0.251 0.235 0.200 0.152 0.135 0.096
Table 1: Asymptotic Distributions of S|z| and A|z| Statistics
29
Asymptotic Distribution of SC Statisticm
Percentile 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.999% 42.488 39.592 38.534 37.694 37.361 35.925 36.051 34.591 31.80495% 37.159 34.956 32.983 32.277 30.902 29.597 29.008 27.773 25.96290% 35.155 32.562 30.722 29.378 28.069 27.176 26.181 24.685 23.35680% 32.624 29.964 28.071 26.673 25.329 24.182 22.935 21.621 20.17570% 30.932 28.330 26.359 24.906 23.421 22.132 20.932 19.423 18.12560% 29.689 26.991 24.908 23.416 21.858 20.579 19.179 17.901 16.55050% 28.540 25.798 23.648 22.111 20.626 19.136 17.708 16.501 15.24140% 27.482 24.522 22.468 20.900 19.254 17.672 16.440 15.111 13.82930% 26.508 23.385 21.197 19.510 17.873 16.413 15.058 13.685 12.47720% 25.351 22.174 19.836 18.193 16.514 14.927 13.582 12.250 10.89810% 23.731 20.400 18.067 16.333 14.568 13.061 11.741 10.480 8.9875% 22.376 18.902 16.821 14.879 13.338 11.876 10.383 9.037 7.8521% 20.394 16.833 14.737 12.807 11.208 9.884 8.275 6.929 5.905
Asymptotic Distribution of AC Statisticm
Percentile 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.999% 16.342 18.610 20.265 22.301 24.658 26.534 27.422 28.202 27.98795% 15.298 16.576 17.794 18.927 19.924 20.838 21.866 22.084 22.53390% 14.762 15.734 16.456 17.287 18.022 18.629 18.996 19.612 19.96580% 14.099 14.648 15.143 15.477 16.014 16.449 16.761 16.975 17.09670% 13.672 14.029 14.209 14.439 14.705 14.937 15.107 15.111 15.29060% 13.295 13.393 13.478 13.566 13.597 13.552 13.629 13.790 13.77250% 12.955 12.873 12.847 12.820 12.666 12.591 12.534 12.514 12.54740% 12.623 12.388 12.178 12.014 11.722 11.471 11.290 11.362 11.31130% 12.262 11.851 11.535 11.227 10.909 10.576 10.364 10.166 10.07320% 11.828 11.264 10.837 10.423 9.991 9.555 9.288 8.965 8.78610% 11.298 10.483 9.871 9.330 8.746 8.297 7.860 7.558 7.1435% 10.840 9.895 9.120 8.436 7.849 7.358 6.750 6.380 5.9671% 10.030 8.791 7.993 7.279 6.363 5.659 5.069 4.629 4.382
Table 2: Asymptotic Distribution of SC and AC Statistics (13 autocontours)
30
Asymptotic Distribution of SL Statisticm
percentile 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.999% 26.825 25.069 23.492 23.010 22.057 20.621 19.801 19.370 17.75695% 22.719 21.121 19.941 18.372 17.199 16.042 15.645 14.470 13.63190% 20.867 19.109 17.489 16.318 15.203 14.241 13.301 12.552 11.63680% 18.859 16.979 15.273 14.045 12.859 11.973 11.213 10.267 9.32570% 17.452 15.505 13.917 12.671 11.665 10.585 9.708 8.788 7.94960% 16.389 14.279 12.920 11.587 10.492 9.537 8.610 7.765 6.92750% 14.810 12.425 11.052 9.796 8.827 7.916 6.830 6.071 5.19240% 15.534 13.300 11.910 10.616 9.576 8.678 7.698 6.917 6.06830% 13.970 11.675 10.164 8.891 7.999 6.948 6.048 5.213 4.42320% 13.066 10.697 9.249 8.023 7.039 6.054 5.113 4.388 3.64610% 11.896 9.535 8.158 7.011 5.863 4.966 4.213 3.484 2.7435% 11.145 8.544 7.070 6.093 5.099 4.286 3.492 2.786 2.1871% 9.583 7.267 5.857 4.821 3.950 3.173 2.585 1.907 1.510
Asymptotic Distribution of AL Statisticm
percentile 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.999% 7.260 8.419 9.620 10.837 12.021 12.943 13.509 14.458 14.62995% 6.507 7.336 7.948 8.396 9.045 9.636 10.116 10.569 10.91390% 6.122 6.673 7.034 7.503 7.961 8.313 8.562 8.929 9.22080% 5.705 6.038 6.259 6.452 6.643 6.822 6.995 7.104 7.30970% 5.383 5.583 5.669 5.733 5.820 5.868 5.929 5.977 6.04060% 5.164 5.208 5.200 5.198 5.212 5.158 5.191 5.122 5.12850% 4.742 4.604 4.468 4.361 4.224 4.021 3.885 3.817 3.76740% 4.951 4.890 4.831 4.770 4.663 4.570 4.506 4.452 4.43830% 4.535 4.316 4.123 3.922 3.717 3.476 3.324 3.174 3.08820% 4.294 3.988 3.712 3.418 3.174 2.973 2.715 2.584 2.48010% 4.017 3.539 3.189 2.922 2.602 2.320 2.086 1.927 1.7465% 3.752 3.237 2.867 2.508 2.246 1.921 1.695 1.459 1.2921% 3.337 2.681 2.222 1.875 1.621 1.388 1.150 0.960 0.777
Table 3: Asymptotic Distribution of SL and AL Statistics (five lags)
31
Tables: Size and Power of the Tests
I. Small Sample (T ) and Small Subsample Window (m)
T=150, R=90,P=60 S1,1|z| S1,2
|z| S1,3|z| S1,4
|z| S1,5|z| S1,6
|z| S1,7|z| S1,8
|z| S1,9|z| S1,10
|z| S1,11|z| S1,12
|z| S1,13|z|
fixed 0.042 0.037 0.033 0.046 0.049 0.037 0.04 0.051 0.045 0.045 0.048 0.036 0.039rolling 0.028 0.04 0.041 0.041 0.038 0.042 0.04 0.04 0.041 0.046 0.044 0.036 0.038
recursive 0.029 0.041 0.042 0.04 0.039 0.039 0.038 0.039 0.044 0.042 0.039 0.035 0.032
A1,1|z| A1,2
|z| A1,3|z| A1,4
|z| A1,5|z| A1,6
|z| A1,7|z| A1,8
|z| A1,9|z| A1,10
|z| A1,11|z| A1,12
|z| A1,13|z|
fixed 0.047 0.048 0.049 0.056 0.055 0.058 0.049 0.059 0.049 0.057 0.052 0.055 0.057rolling 0.032 0.038 0.048 0.043 0.041 0.046 0.046 0.045 0.04 0.046 0.048 0.046 0.031
recursive 0.024 0.037 0.037 0.043 0.043 0.038 0.043 0.052 0.043 0.044 0.042 0.035 0.033
S1,13C A1,13
C S2,7L S3,7
L S4,7L S5,7
L A2,7L A3,7
L A4,7L A5,7
L
fixed 0.059 0.051 0.06 0.06 0.052 0.045 0.063 0.063 0.058 0.055rolling 0.038 0.043 0.045 0.052 0.042 0.042 0.051 0.05 0.045 0.045
recursive 0.033 0.043 0.044 0.055 0.053 0.052 0.052 0.058 0.039 0.033
S1,7|z| S2,7
|z| S3,7|z| S4,7
|z| S5,7|z| A1,7
|z| A2,7|z| A3,7
|z| A4,7|z| A5,7
|z|fixed 0.04 0.038 0.042 0.04 0.038 0.049 0.053 0.054 0.051 0.039
rolling 0.04 0.033 0.032 0.033 0.034 0.036 0.038 0.039 0.04 0.032recursive 0.038 0.037 0.031 0.035 0.034 0.043 0.039 0.04 0.05 0.037
Notation: 13 autocontours C = [0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99].
In Sk,αi|z| and Ak,αi|z| , lag k and autocontour αi are fixed.
In Sk,αC and Ak,αC , k is fixed and α is the total number of autocontours considered.
In Sk,αiL and Ak,αiL , up to k lags are considered and αi is a fixed autocontour.
Table 4: Size of the statistics:T=150 R=90 P=T-R=60 m = 1/3 (nominal size 5%)
32
II. Large Sample (T ) and Large Subsample Window (m)
T=750, R=450,P=300 S1,1|z| S1,2
|z| S1,3|z| S1,4
|z| S1,5|z| S1,6
|z| S1,7|z| S1,8
|z| S1,9|z| S1,10
|z| S1,11|z| S1,12
|z| S1,13|z|
fixed 0.048 0.040 0.044 0.049 0.046 0.051 0.052 0.048 0.052 0.04 0.042 0.05 0.032rolling 0.049 0.041 0.056 0.051 0.048 0.041 0.041 0.045 0.057 0.056 0.049 0.038 0.039
recursive 0.046 0.048 0.045 0.044 0.051 0.048 0.044 0.045 0.047 0.057 0.058 0.039 0.04
A1,1|z| A1,2
|z| A1,3|z| A1,4
|z| A1,5|z| A1,6
|z| A1,7|z| A1,8
|z| A1,9|z| A1,10
|z| A1,11|z| A1,12
|z| A1,13|z|
fixed 0.049 0.045 0.049 0.051 0.046 0.049 0.051 0.043 0.049 0.052 0.043 0.047 0.049rolling 0.048 0.042 0.046 0.055 0.044 0.047 0.043 0.044 0.044 0.043 0.049 0.044 0.048
recursive 0.04 0.043 0.041 0.063 0.065 0.045 0.053 0.045 0.044 0.051 0.042 0.044 0.043
S1,13C A1,13
C S2,7L S3,7
L S4,7L S5,7
L A2,7L A3,7
L A4,7L A5,7
L
fixed 0.045 0.049 0.047 0.044 0.053 0.05 0.049 0.042 0.046 0.051rolling 0.045 0.044 0.043 0.064 0.063 0.044 0.048 0.041 0.044 0.05
recursive 0.043 0.047 0.045 0.057 0.052 0.048 0.048 0.048 0.051 0.048
S1,7|z| S2,7
|z| S3,7|z| S4,7
|z| S5,7|z| A1,7
|z| A2,7|z| A3,7
|z| A4,7|z| A5,7
|z|fixed 0.052 0.053 0.042 0.042 0.043 0.051 0.048 0.042 0.054 0.049
rolling 0.041 0.048 0.043 0.039 0.04 0.043 0.043 0.047 0.049 0.043recursive 0.044 0.051 0.041 0.038 0.041 0.053 0.054 0.043 0.045 0.048
Notation: 13 autocontours C = [0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99].
In Sk,αi|z| and Ak,αi|z| , lag k and autocontour αi are fixed.
In Sk,αC and Ak,αC , k is fixed and α is the total number of autocontours considered.
In Sk,αiL and Ak,αiL , up to k lags are considered and αi is a fixed autocontour.
Table 5: Size of the statistics:T=750 R=450 P=T-R=300 m = 2/3 (nominal size 5%)
III. Small Sample (T ), Small Subsample Window (m), and P/R = 1/4
T=120, R=96, P/R = 1/4 S1,1|z| S1,2
|z| S1,3|z| S1,4
|z| S1,5|z| S1,6
|z| S1,7|z| S1,8
|z| S1,9|z| S1,10
|z| S1,11|z| S1,12
|z| S1,13|z|
fixed 0.015 0.023 0.024 0.032 0.038 0.034 0.032 0.033 0.037 0.031 0.025 0.023 0.016rolling 0.012 0.019 0.016 0.028 0.031 0.035 0.033 0.026 0.034 0.032 0.023 0.018 0.013
recursive 0.014 0.021 0.017 0.03 0.029 0.032 0.031 0.028 0.03 0.03 0.015 0.016 0.015
A1,1|z| A1,2
|z| A1,3|z| A1,4
|z| A1,5|z| A1,6
|z| A1,7|z| A1,8
|z| A1,9|z| A1,10
|z| A1,11|z| A1,12
|z| A1,13|z|
fixed 0.029 0.032 0.04 0.047 0.043 0.043 0.044 0.044 0.046 0.052 0.029 0.032 0.021rolling 0.023 0.025 0.025 0.035 0.045 0.045 0.04 0.045 0.045 0.041 0.027 0.025 0.021
recursive 0.021 0.028 0.024 0.031 0.043 0.046 0.033 0.042 0.045 0.042 0.027 0.027 0.023
S1,13C A1,13
C S2,7L S3,7
L S4,7L S5,7
L A2,7L A3,7
L A4,7L A5,7
L
fixed 0.038 0.032 0.03 0.034 0.032 0.03 0.045 0.041 0.042 0.042rolling 0.032 0.033 0.032 0.037 0.035 0.036 0.044 0.042 0.051 0.046
recursive 0.029 0.031 0.035 0.033 0.034 0.032 0.045 0.044 0.039 0.042
S1,7|z| S2,7
|z| S3,7|z| S4,7
|z| S5,7|z| A1,7
|z| A2,7|z| A3,7
|z| A4,7|z| A5,7
|z|fixed 0.032 0.033 0.037 0.032 0.034 0.044 0.043 0.044 0.042 0.039
rolling 0.033 0.032 0.033 0.034 0.031 0.041 0.042 0.04 0.039 0.038recursive 0.031 0.03 0.029 0.032 0.029 0.033 0.032 0.031 0.034 0.031
Notation: 13 autocontours C = [0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99].
In Sk,αi|z| and Ak,αi|z| , lag k and autocontour αi are fixed.
In Sk,αC and Ak,αC , k is fixed and α is the total number of autocontours considered.
In Sk,αiL and Ak,αiL , up to k lags are considered and αi is a fixed autocontour.
Table 6: Size of the statistics: T=120, R=96, P=24, P/R = 1/4 m = 1/3 (nominal size5%)
33
IV. Small Sample (T ), Small Subsample Window (m), and P/R = 1/2
T=144, R=96, P/R = 1/2 S1,1|z| S1,2
|z| S1,3|z| S1,4
|z| S1,5|z| S1,6
|z| S1,7|z| S1,8
|z| S1,9|z| S1,10
|z| S1,11|z| S1,12
|z| S1,13|z|
fixed 0.022 0.029 0.032 0.037 0.041 0.042 0.034 0.039 0.04 0.044 0.027 0.026 0.023rolling 0.019 0.026 0.027 0.035 0.039 0.041 0.039 0.032 0.041 0.046 0.038 0.032 0.02
recursive 0.021 0.029 0.026 0.032 0.034 0.035 0.035 0.033 0.042 0.043 0.034 0.021 0.023
A1,1|z| A1,2
|z| A1,3|z| A1,4
|z| A1,5|z| A1,6
|z| A1,7|z| A1,8
|z| A1,9|z| A1,10
|z| A1,11|z| A1,12
|z| A1,13|z|
fixed 0.034 0.035 0.04 0.043 0.048 0.056 0.043 0.044 0.04 0.047 0.036 0.033 0.027rolling 0.032 0.034 0.033 0.045 0.043 0.041 0.041 0.046 0.043 0.044 0.033 0.032 0.028
recursive 0.031 0.032 0.03 0.046 0.042 0.042 0.04 0.045 0.047 0.045 0.034 0.03 0.032
S1,13C A1,13
C S2,7L S3,7
L S4,7L S5,7
L A2,7L A3,7
L A4,7L A5,7
L
fixed 0.037 0.044 0.037 0.036 0.038 0.038 0.043 0.042 0.046 0.046rolling 0.034 0.042 0.039 0.037 0.037 0.036 0.041 0.045 0.042 0.044
recursive 0.031 0.039 0.032 0.033 0.034 0.034 0.042 0.043 0.044 0.042
S1,7|z| S2,7
|z| S3,7|z| S4,7
|z| S5,7|z| A1,7
|z| A2,7|z| A3,7
|z| A4,7|z| A5,7
|z|fixed 0.035 0.04 0.038 0.039 0.038 0.043 0.05 0.044 0.049 0.043
rolling 0.039 0.036 0.037 0.037 0.039 0.041 0.048 0.041 0.039 0.04recursive 0.035 0.039 0.033 0.038 0.036 0.04 0.041 0.039 0.042 0.039
Notation: 13 autocontours C = [0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99].
In Sk,αi|z| and Ak,αi|z| , lag k and autocontour αi are fixed.
In Sk,αC and Ak,αC , k is fixed and α is the total number of autocontours considered.
In Sk,αiL and Ak,αiL , up to k lags are considered and αi is a fixed autocontour.
Table 7: Size of the statistics: T=144, R=96, P=48, P/R = 1/2 m = 1/3 (nominal size5%)
V. Small Sample (T ), Small Subsample Window (m), and P/R = 3/4
T=168, R=96, P/R = 3/4 S1,1|z| S1,2
|z| S1,3|z| S1,4
|z| S1,5|z| S1,6
|z| S1,7|z| S1,8
|z| S1,9|z| S1,10
|z| S1,11|z| S1,12
|z| S1,13|z|
fixed 0.032 0.041 0.04 0.042 0.043 0.042 0.041 0.046 0.046 0.046 0.043 0.032 0.039rolling 0.031 0.04 0.044 0.04 0.04 0.046 0.046 0.049 0.051 0.046 0.051 0.031 0.038
recursive 0.029 0.042 0.043 0.053 0.052 0.049 0.043 0.043 0.042 0.042 0.039 0.036 0.035
A1,1|z| A1,2
|z| A1,3|z| A1,4
|z| A1,5|z| A1,6
|z| A1,7|z| A1,8
|z| A1,9|z| A1,10
|z| A1,11|z| A1,12
|z| A1,13|z|
fixed 0.042 0.046 0.047 0.043 0.042 0.043 0.041 0.48 0.052 0.045 0.041 0.038 0.038rolling 0.037 0.039 0.04 0.045 0.045 0.047 0.046 0.047 0.051 0.042 0.043 0.037 0.039
recursive 0.038 0.042 0.037 0.048 0.049 0.047 0.042 0.045 0.049 0.44 0.039 0.036 0.038
S1,13C A1,13
C S2,7L S3,7
L S4,7L S5,7
L A2,7L A3,7
L A4,7L A5,7
L
fixed 0.04 0.044 0.043 0.044 0.043 0.045 0.046 0.046 0.045 0.047rolling 0.04 0.052 0.043 0.05 0.044 0.049 0.049 0.044 0.045 0.45
recursive 0.042 0.052 0.046 0.043 0.042 0.04 0.043 0.045 0.044 0.045
S1,7|z| S2,7
|z| S3,7|z| S4,7
|z| S5,7|z| A1,7
|z| A2,7|z| A3,7
|z| A4,7|z| A5,7
|z|fixed 0.041 0.047 0.046 0.044 0.045 0.041 0.041 0.045 0.043 0.045
rolling 0.046 0.04 0.042 0.043 0.044 0.046 0.048 0.046 0.045 0.043recursive 0.043 0.041 0.042 0.046 0.044 0.042 0.043 0.045 0.044 0.042
Notation: 13 autocontours C = [0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99].
In Sk,αi|z| and Ak,αi|z| , lag k and autocontour αi are fixed.
In Sk,αC and Ak,αC , k is fixed and α is the total number of autocontours considered.
In Sk,αiL and Ak,αiL , up to k lags are considered and αi is a fixed autocontour.
Table 8: Size of the statistics: T=168, R=96, P=72, P/R = 3/4 m = 1/3 (nominal size5%)
34
Fixed Scheme Sl,1|z| Sl,2|z| Sl,3|z| Sl,4|z| Sl,5|z| Sl,6|z| Sl,7|z| Sl,8|z| Sl,9|z| Sl,10|z| Sl,11
|z| Sl,12|z| Sl,13
|z|
l = 1 0.14 0.45 0.765 0.956 0.979 0.987 0.995 0.992 0.987 0.976 0.938 0.867 0.589l = 2 0.14 0.464 0.789 0.955 0.981 0.99 0.993 0.993 0.988 0.974 0.943 0.864 0.593l = 3 0.14 0.419 0.792 0.965 0.985 0.997 0.997 0.994 0.985 0.971 0.937 0.871 0.594l = 4 0.11 0.439 0.797 0.953 0.986 0.993 0.994 0.994 0.99 0.979 0.938 0.866 0.595l = 5 0.12 0.453 0.796 0.952 0.984 0.992 0.994 0.993 0.989 0.971 0.938 0.867 0.598
Al,1|z| Al,2|z| Al,3|z| Al,4|z| Al,5|z| Al,6|z| Al,7|z| Al,8|z| Al,9|z| Al,10|z| Al,11
|z| Al,12|z| Al,13
|z|l = 1 0.08 0.244 0.698 0.898 0.948 0.969 0.981 0.978 0.968 0.945 0.904 0.812 0.588l = 2 0.09 0.277 0.741 0.896 0.952 0.965 0.979 0.983 0.972 0.947 0.898 0.815 0.585l = 3 0.07 0.276 0.72 0.893 0.941 0.969 0.98 0.984 0.967 0.943 0.906 0.815 0.588l = 4 0.07 0.319 0.702 0.887 0.957 0.971 0.979 0.981 0.971 0.944 0.904 0.817 0.586l = 5 0.08 0.339 0.722 0.906 0.951 0.968 0.985 0.976 0.972 0.944 0.903 0.816 0.584
Sl,13C Al,13
C
l = 1 0.961 0.921l = 2 0.965 0.924l = 3 0.967 0.923l = 4 0.963 0.921l = 5 0.965 0.921
S2,7L S3,7
L S4,7L S5,7
L A2,7L A3,7
L A4,7L A5,7
L
C = 7 0.989 0.985 0.976 0.964 0.98 0.969 0.956 0.942
Rolling Scheme Sl,1|z| Sl,2|z| Sl,3|z| Sl,4|z| Sl,5|z| Sl,6|z| Sl,7|z| Sl,8|z| Sl,9|z| Sl,10|z| Sl,11
|z| Sl,12|z| Sl,13
|z|
l = 1 0.18 0.12 0.308 0.416 0.459 0.394 0.42 0.264 0.21 0.08 0.119 0.092 0.08l = 2 0.11 0.11 0.295 0.324 0.323 0.431 0.433 0.333 0.193 0.084 0.123 0.093 0.08l = 3 0.11 0.11 0.258 0.402 0.386 0.39 0.472 0.325 0.207 0.096 0.123 0.094 0.09l = 4 0.13 0.11 0.289 0.439 0.41 0.413 0.378 0.325 0.173 0.093 0.121 0.092 0.082l = 5 0.16 0.11 0.262 0.445 0.418 0.423 0.438 0.318 0.171 0.091 0.121 0.092 0.083
Al,1|z| Al,2|z| Al,3|z| Al,4|z| Al,5|z| Al,6|z| Al,7|z| Al,8|z| Al,9|z| Al,10|z| Al,11
|z| Al,12|z| Al,13
|z|l = 1 0.25 0.318 0.559 0.783 0.738 0.710 0.534 0.634 0.418 0.18 0.21 0.102 0.091l = 2 0.22 0.292 0.527 0.733 0.71 0.714 0.56 0.621 0.438 0.17 0.216 0.100 0.092l = 3 0.24 0.275 0.521 0.768 0.756 0.744 0.499 0.638 0.442 0.155 0.224 0.103 0.092l = 4 0.25 0.28 0.519 0.806 0.775 0.759 0.611 0.638 0.421 0.133 0.21 0.101 0.092l = 5 0.25 0.305 0.516 0.797 0.781 0.769 0.533 0.626 0.392 0.121 0.24 0.101 0.093
Sl,13,C Al,13
C
l = 1 0.638 0.749l = 2 0.521 0.733l = 3 0.619 0.801l = 4 0.641 0.822l = 5 0.637 0.803
S2,7L S3,7
L S4,7L S5,7
L A2,7L A3,7
L A4,7L A5,7
L
C = 7 0.414 0.419 0.408 0.432 0.563 0.59 0.578 0.532
Notes: 13 autocontours C = [0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99].
Sl,7|z| ,Al,7|z| for l = 1, 2, ...5; 7 refers to the 50% autocontour.
Sl,7L ,Al,7L stacking lags up to l = 2, ....5 and considering the 50% autocontour.
Sl,13C and Al,13
C stacking all 13 autocontours for one lag l = 1, 2, 3, 4, 5.1000 Monte Carlo replications and 500 bootstrap samples.T=650, R=350, P=300, m = 1/3, and break point at R + τP for τ = 1/3
Table 9: Power for DGP1 under Fixed and Rolling Schemes
35
Fixed Scheme Sl,1|z| Sl,2|z| Sl,3|z| Sl,4|z| Sl,5|z| Sl,6|z| Sl,7|z| Sl,8|z| Sl,9|z| Sl,10|z| Sl,11
|z| Sl,12|z| Sl,13
|z|
l = 1 0.292 0.271 0.191 0.084 0.073 0.109 0.281 0.278 0.391 0.512 0.638 0.638 0.518l = 2 0.26 0.239 0.171 0.079 0.075 0.106 0.282 0.292 0.418 0.518 0.637 0.639 0.52l = 3 0.285 0.265 0.175 0.089 0.075 0.12 0.283 0.29 0.404 0.53 0.635 0.639 0.523l = 4 0.311 0.26 0.162 0.088 0.063 0.096 0.282 0.284 0.399 0.512 0.624 0.636 0.526l = 5 0.273 0.253 0.158 0.068 0.068 0.108 0.282 0.288 0.408 0.539 0.645 0.638 0.531
Al,1|z| Al,2|z| Al,3|z| Al,4|z| Al,5|z| Al,6|z| Al,7|z| Al,8|z| Al,9|z| Al,10|z| Al,11
|z| Al,12|z| Al,13
|z|l = 1 0.345 0.242 0.162 0.09 0.078 0.127 0.275 0.271 0.361 0.472 0.561 0.604 0.527l = 2 0.327 0.225 0.16 0.081 0.077 0.12 0.269 0.277 0.377 0.467 0.56 0.606 0.522l = 3 0.347 0.239 0.167 0.089 0.087 0.123 0.272 0.286 0.374 0.468 0.558 0.6 0.527l = 4 0.368 0.246 0.159 0.09 0.069 0.111 0.272 0.277 0.345 0.455 0.55 0.592 0.523l = 5 0.317 0.222 0.146 0.086 0.072 0.12 0.273 0.288 0.37 0.469 0.564 0.6 0.523
Sl,13C Al,13
C
l = 1 0.659 0.595l = 2 0.651 0.604l = 3 0.666 0.625l = 4 0.644 0.625l = 5 0.652 0.611
S2,7L S3,7
L S4,7L S5,7
L A2,7L A3,7
L A4,7L A5,7
L
C = 7 0.286 0.285 0.281 0.285 0.271 0.273 0.276 0.275
Rolling Scheme Sl,1|z| Sl,2|z| Sl,3|z| Sl,4|z| Sl,5|z| Sl,6|z| Sl,7|z| Sl,8|z| Sl,9|z| Sl,10|z| Sl,11
|z| Sl,12|z| Sl,13
|z|
l = 1 0.116 0.104 0.088 0.063 0.05 0.056 0.112 0.11 0.171 0.341 0.302 0.264 0.261l = 2 0.114 0.09 0.086 0.063 0.054 0.051 0.118 0.113 0.172 0.318 0.296 0.264 0.262l = 3 0.107 0.094 0.076 0.065 0.06 0.053 0.108 0.126 0.189 0.335 0.28 0.273 0.266l = 4 0.113 0.104 0.072 0.054 0.058 0.049 0.175 0.114 0.167 0.329 0.287 0.266 0.267l = 5 0.109 0.093 0.08 0.053 0.05 0.053 0.16 0.121 0.184 0.337 0.274 0.265 0.267
Al,1|z| Al,2|z| Al,3|z| Al,4|z| Al,5|z| Al,6|z| Al,7|z| Al,8|z| Al,9|z| Al,10|z| Al,11
|z| Al,12|z| Al,13
|z|l = 1 0.179 0.129 0.099 0.078 0.066 0.075 0.19 0.151 0.204 0.36 0.344 0.35 0.378l = 2 0.169 0.112 0.082 0.075 0.066 0.076 0.18 0.145 0.205 0.386 0.345 0.343 0.379l = 3 0.175 0.139 0.092 0.07 0.067 0.077 0.196 0.159 0.228 0.378 0.352 0.348 0.378l = 4 0.183 0.143 0.083 0.063 0.066 0.077 0.192 0.144 0.201 0.358 0.342 0.352 0.375l = 5 0.159 0.129 0.09 0.063 0.066 0.065 0.191 0.151 0.208 0.371 0.347 0.35 0.377
Sl,13C Al,13
C
l = 1 0.409 0.414l = 2 0.363 0.394l = 3 0.381 0.427l = 4 0.398 0.431l = 5 0.306 0.318
S2,7L S3,7
L S4,7L S5,7
L A2,7L A3,7
L A4,7L A5,7
L
C = 7 0.168 0.115 0.113 0.112 0.129 0.189 0.191 0.182
Notes: 13 autocontours C = [0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99].
Sl,7|z| ,Al,7|z| for l = 1, 2, ...5; 7 refers to the 50% autocontour.
Sl,7L ,Al,7L stacking lags up to l = 2, ....5 and considering the 50% autocontour.
Sl,13C and Al,13
C stacking all 13 autocontours for one lag l = 1, 2, 3, 4, 5.1000 Monte Carlo replications and 500 bootstrap samples.T=650, R=350, P=300, m = 1/3, and break point at R + τP for τ = 1/3
Table 10: Power for DGP2 under Fixed and Rolling Schemes
36
Fixed Scheme Sl,1|z| Sl,2|z| Sl,3|z| Sl,4|z| Sl,5|z| Sl,6|z| Sl,7|z| Sl,8|z| Sl,9|z| Sl,10|z| Sl,11
|z| Sl,12|z| Sl,13
|z|
l = 1 0.95 0.998 0.999 0.999 0.999 1 0.97 0.985 0.961 0.849 0.449 0.33 0.29l = 2 0.96 0.998 0.999 1 0.999 0.997 0.982 0.981 0.96 0.859 0.444 0.16 0.19l = 3 0.97 0.997 0.998 1 0.999 0.998 0.979 0.982 0.956 0.85 0.436 0.35 0.31l = 4 0.96 0.997 0.998 1 0.998 0.998 0.98 0.984 0.965 0.853 0.446 0.36 0.13l = 5 0.96 0.997 1 0.999 1 0.999 0.982 0.983 0.96 0.854 0.447 0.2 0.23
Al,1|z| Al,2|z| Al,3|z| Al,4|z| Al,5|z| Al,6|z| Al,7|z| Al,8|z| Al,9|z| Al,10|z| Al,11
|z| Al,12|z| Al,13
|z|l = 1 0.93 0.951 1 0.999 0.999 0.995 0.95 0.973 0.914 0.475 0.38 0.25 0.16l = 2 0.931 0.967 0.998 1 0.999 0.994 0.952 0.974 0.896 0.518 0.34 0.22 0.17l = 3 0.937 0.98 0.998 0.999 0.997 0.995 0.95 0.973 0.883 0.555 0.36 0.24 0.19l = 4 0.935 0.96 1 0.998 0.997 0.994 0.95 0.968 0.908 0.487 0.31 0.32 0.22l = 5 0.939 0.95 0.999 0.998 0.997 0.995 0.95 0.962 0.875 0.545 0.31 0.12 0.23
Sl,13C Al,13
C
l = 1 1 0.97l = 2 1 0.98l = 3 0.999 0.97l = 4 1 0.999l = 5 1 1
S2,7L S3,7
L S4,7L S5,7
L A2,7L A3,7
L A4,7L A5,7
L
C = 7 0.98 0.98 0.98 0.98 0.97 0.969 0.97 0.97
Rolling Scheme Sl,1|z| Sl,2|z| Sl,3|z| Sl,4|z| Sl,5|z| Sl,6|z| Sl,7|z| Sl,8|z| Sl,9|z| Sl,10|z| Sl,11
|z| Sl,12|z| Sl,13
|z|
l = 1 0.46 0.461 0.395 0.184 0.26 0.329 0.396 0.457 0.401 0.134 0.212 0.092 0.08l = 2 0.358 0.32 0.338 0.381 0.421 0.475 0.511 0.508 0.39 0.121 0.241 0.093 0.08l = 3 0.437 0.398 0.389 0.442 0.465 0.524 0.524 0.518 0.385 0.099 0.223 0.094 0.09l = 4 0.445 0.425 0.444 0.481 0.502 0.544 0.54 0.512 0.345 0.095 0.232 0.092 0.082l = 5 0.448 0.409 0.425 0.468 0.511 0.54 0.546 0.506 0.361 0.075 0.232 0.092 0.083
Al,1|z| Al,2|z| Al,3|z| Al,4|z| Al,5|z| Al,6|z| Al,7|z| Al,8|z| Al,9|z| Al,10|z| Al,11
|z| Al,12|z| Al,13
|z|l = 1 0.189 0.219 0.332 0.473 0.584 0.678 0.726 0.739 0.706 0.395 0.163 0.082 0.06l = 2 0.585 0.628 0.686 0.788 0.808 0.834 0.826 0.815 0.716 0.356 0.171 0.081 0.06l = 3 0.68 0.711 0.728 0.796 0.832 0.839 0.832 0.811 0.7 0.334 0.163 0.081 0.07l = 4 0.684 0.733 0.765 0.832 0.851 0.861 0.846 0.815 0.698 0.308 0.162 0.081 0.062l = 5 0.674 0.717 0.762 0.82 0.846 0.86 0.848 0.814 0.669 0.291 0.161 0.081 0.063
Sl,13C Al,13
C
l = 1 0.57 0.817l = 2 0.539 0.845l = 3 0.53 0.814l = 4 0.532 0.78l = 5 0.534 0.782
S2,7L S3,7
L S4,7L S5,7
L A2,7L A3,7
L A4,7L A5,7
L
0.477 0.496 0.48 0.421 0.728 0.75 0.749 0.703
Notes: 13 autocontours C = [0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99].
Sl,7|z| ,Al,7|z| for l = 1, 2, ...5; 7 refers to the 50% autocontour.
Sl,7L ,Al,7L stacking lags up to l = 2, ....5 and considering the 50% autocontour.
Sl,13C and Al,13
C stacking all 13 autocontours for one lag l = 1, 2, 3, 4, 5.1000 Monte Carlo replications and 500 bootstrap samples.T=650, R=350, P=300, m = 1/3, and break point at R + τP for τ = 1/3
Table 11: Power for DGP3 under Fixed and Rolling Schemes
37
Fixed Scheme Sl,1|z| Sl,2|z| Sl,3|z| Sl,4|z| Sl,5|z| Sl,6|z| Sl,7|z| Sl,8|z| Sl,9|z| Sl,10|z| Sl,11
|z| Sl,12|z| Sl,13
|z|
l = 1 1 1 0.999 1 1 1 0.999 0.999 0.989 0.984 0.969 0.93 0.79l = 2 0.999 1 0.999 0.996 1 1 1 1 0.98 0.983 0.967 0.901 0.78l = 3 0.998 1 1 1 1 0.999 1 1 0.996 0.98 0.965 0.912 0.781l = 4 1 0.999 1 1 0.998 1 0.999 1 0.99 0.98 0.962 0.923 0.744l = 5 1 0.996 1 1 1 1 1 0.998 1 0.98 0.966 0.917 0.775
Al,1|z| Al,2|z| Al,3|z| Al,4|z| Al,5|z| Al,6|z| Al,7|z| Al,8|z| Al,9|z| Al,10|z| Al,11
|z| Al,12|z| Al,13
|z|l = 1 0.989 1 0.995 1 1 1 0.98 0.998 0.989 0.96 0.914 0.812 0.639l = 2 0.989 1 0.995 0.995 0.997 1 0.988 0.989 0.986 0.967 0.934 0.815 0.633l = 3 0.99 1 1 1 1 0.997 0.985 0.99 0.986 0.966 0.914 0.815 0.622l = 4 1 0.996 1 1 0.996 0.. 0.989 0.991 0.986 0.967 0.954 0.817 0.644l = 5 1 0.996 1 1 1 1 0.989 0.996 0.986 0.97 0.914 0.816 0.675
Sl,13C Al,13
C
l = 1 1 1l = 2 1 1l = 3 1 1l = 4 1 1l = 5 1 1
S2,7L S3,7
L S4,7L S5,7
L A2,7L A3,7
L A4,7L A5,7
L
C = 7 1 1 1 1 1 1 1 1
Rolling Scheme Sl,1|z| Sl,2|z| Sl,3|z| Sl,4|z| Sl,5|z| Sl,6|z| Sl,7|z| Sl,8|z| Sl,9|z| Sl,10|z| Sl,11
|z| Sl,12|z| Sl,13
|z|
l = 1 0.569 0.461 0.486 0.589 0.696 0.746 0.76 0.729 0.69 0.643 0.481 0.331 0.123l = 2 0.44 0.387 0.358 0.577 0.664 0.73 0.744 0.733 0.703 0.629 0.464 0.332 0.12l = 3 0.541 0.465 0.421 0.528 0.649 0.695 0.705 0.71 0.691 0.624 0.445 0.332 0.127l = 4 0.586 0.514 0.48 0.525 0.639 0.698 0.703 0.694 0.663 0.618 0.447 0.319 0.131l = 5 0.571 0.495 0.484 0.525 0.643 0.724 0.722 0.723 0.678 0.627 0.448 0.318 0.135
Al,1|z| Al,2|z| Al,3|z| Al,4|z| Al,5|z| Al,6|z| Al,7|z| Al,8|z| Al,9|z| Al,10|z| Al,11
|z| Al,12|z| Al,13
|z|l = 1 0.25 0.318 0.559 0.761 0.831 0.861 0.858 0.839 0.819 0.763 0.606 0.461 0.236l = 2 0.22 0.292 0.527 0.739 0.794 0.835 0.857 0.852 0.82 0.767 0.601 0.46 0.242l = 3 0.24 0.275 0.521 0.705 0.789 0.835 0.85 0.854 0.819 0.767 0.609 0.45 0.237l = 4 0.25 0.28 0.519 0.699 0.795 0.833 0.86 0.842 0.8 0.736 0.606 0.443 0.24l = 5 0.25 0.305 0.516 0.704 0.789 0.837 0.846 0.848 0.823 0.75 0.588 0.446 0.242
Sl,13C Al,13
C
l = 1 0.75 0.85l = 2 0.74 0.85l = 3 0.76 0.87l = 4 0.76 0.83l = 5 0.771 0.81
S2,7L S3,7
L S4,7L S5,7
L A2,7L A3,7
L A4,7L A5,7
L
C = 7 0.656 0.558 0.558 0.56 0.801 0.724 0.656 0.605
Notes: 13 autocontours C = [0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99].
Sl,7|z| ,Al,7|z| for l = 1, 2, ...5; 7 refers to the 50% autocontour.
Sl,7L ,Al,7L stacking lags up to l = 2, ....5 and considering the 50% autocontour.
Sl,13C and Al,13
C stacking all 13 autocontours for one lag l = 1, 2, 3, 4, 5.1000 Monte Carlo replications and 500 bootstrap samples.T=650, R=350, P=300, m = 1/3, and break point at R + τP for τ = 1/3
Table 12: Power for DGP4 under Fixed and Rolling Schemes
38
Rolling Scheme τ = 1/2 Sl,1|z| Sl,2|z| Sl,3|z| Sl,4|z| Sl,5|z| Sl,6|z| Sl,7|z| Sl,8|z| Sl,9|z| Sl,10|z| Sl,11
|z| Sl,12|z| Sl,13
|z|
l = 1 0.326 0.322 0.315 0.132 0.198 0.278 0.352 0.425 0.368 0.109 0.183 0.081 0.069l = 2 0.305 0.246 0.286 0.304 0.294 0.413 0.442 0.456 0.348 0.113 0.193 0.079 0.072l = 3 0.318 0.322 0.301 0.365 0.348 0.401 0.463 0.427 0.372 0.078 0.191 0.083 0.079l = 4 0.322 0.316 0.325 0.392 0.456 0.487 0.459 0.464 0.301 0.083 0.192 0.087 0.069l = 5 0.341 0.301 0.297 0.387 0.432 0.445 0.476 0.447 0.351 0.069 0.177 0.072 0.069
Al,1|z| Al,2|z| Al,3|z| Al,4|z| Al,5|z| Al,6|z| Al,7|z| Al,8|z| Al,9|z| Al,10|z| Al,11
|z| Al,12|z| Al,13
|z|l = 1 0.143 0.187 0.216 0.342 0.403 0.503 0.631 0.645 0.614 0.314 0.133 0.064 0.052l = 2 0.273 0.613 0.632 0.723 0.765 0.728 0.668 0.727 0.625 0.322 0.124 0.062 0.046l = 3 0.404 0.647 0.687 0.745 0.794 0.716 0.589 0.724 0.653 0.301 0.116 0.067 0.064l = 4 0.382 0.587 0.618 0.742 0.783 0.786 0.586 0.722 0.622 0.246 0.125 0.065 0.053l = 5 0.313 0.578 0.628 0.765 0.784 0.785 0.658 0.729 0.635 0.204 0.119 0.065 0.052
S13,lC A13,1
C
l = 1 0.513 0.653l = 2 0.452 0.654l = 3 0.447 0.649l = 4 0.501 0.639l = 5 0.472 0.624
S2,7C S3,7
C S4,7C S5,7
C A2,7C A3,7
C A4,7C A5,7
C
C = 7 0.358 0.442 0.436 0.397 0.694 0.672 0.576 0.542
Rolling Scheme τ = 2/3 Sl,1|z| Sl,2|z| Sl,3|z| Sl,4|z| Sl,5|z| Sl,6|z| Sl,7|z| Sl,8|z| Sl,9|z| Sl,10|z| Sl,11
|z| Sl,12|z| Sl,13
|z|l = 1 0.221 0.206 0.221 0.106 0.148 0.214 0.268 0.397 0.261 0.096 0.132 0.067 0.056l = 2 0.215 0.194 0.214 0.228 0.226 0.331 0.345 0.402 0.279 0.101 0.152 0.071 0.061l = 3 0.242 0.247 0.205 0.204 0.215 0.347 0.344 0.401 0.244 0.069 0.144 0.068 0.053l = 4 0.212 0.272 0.232 0.224 0.222 0.362 0.353 0.398 0.235 0.073 0.132 0.068 0.058l = 5 0.253 0.265 0.247 0.252 0.253 0.343 0.385 0.408 0.245 0.062 0.147 0.063 0.057
Al,1|z| Al,2|z| Al,3|z| Al,4|z| Al,5|z| Al,6|z| Al,7|z| Al,8|z| Al,9|z| Al,10|z| Al,11
|z| Al,12|z| Al,13
|z|l = 1 0.112 0.154 0.164 0.213 0.361 0.453 0.538 0.597 0.526 0.223 0.103 0.056 0.042l = 2 0.214 0.527 0.487 0.583 0.644 0.682 0.523 0.623 0.535 0.213 0.102 0.054 0.041l = 3 0.253 0.562 0.524 0.572 0.634 0.672 0.474 0.643 0.534 0.223 0.109 0.057 0.042l = 4 0.302 0.475 0.516 0.602 0.657 0.685 0.454 0.638 0.547 0.156 0.105 0.058 0.042l = 5 0.268 0.467 0.537 0.586 0.635 0.674 0.464 0.648 0.518 0.157 0.103 0.057 0.043
S13,lC A13,1
C
l = 1 0.423 0.544l = 2 0.413 0.538l = 3 0.409 0.563l = 4 0.427 0.557l = 5 0.418 0.556
S2,7C S3,7
C S4,7C S5,7
C A2,7C A3,7
C A4,7C A5,7
C
C = 7 0.305 0.334 0.354 0.375 0.514 0.523 0.517 0.498
Notes: 13 autocontours C = [0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99].
Sl,7|z| ,Al,7|z| for l = 1, 2, ...5; 7 refers to the 50% autocontour.
Sl,7L ,Al,7L stacking lags up to l = 2, ....5 and considering the 50% autocontour.
Sl,13C and Al,13
C stacking all 13 autocontours for one lag l = 1, 2, 3, 4, 5.1000 Monte Carlo replications and 500 bootstrap samples.T=650, R=350, P=300, m = 1/3, and break point at R + τP for τ = 1/2, 2/3
Table 13: Power for DGP3 under Rolling Scheme τ = 1/2 and 2/3
39
Tables: Phillips CurveFixed Scheme Sl,1|z| Sl,2|z| Sl,3|z| Sl,4|z| Sl,5|z| Sl,6|z| Sl,7|z| Sl,8|z| Sl,9|z| Sl,10
|z| Sl,11|z| Sl,12
|z| Sl,13|z|
l = 1 0.017 0.529 0.523 0.471 0.268 0.002 0.008 0.045 0.136 0.091 0.085 0.031 0.0001l = 2 0.015 0.285 0.55 0.434 0.481 0.002 0.005 0.041 0.157 0.086 0.09 0.027 0.0001l = 3 0.028 0.584 0.614 0.651 0.419 0.006 0.005 0.03 0.049 0.088 0.073 0.013 0.0001l = 4 0.013 0.104 0.807 0.285 0.209 0.009 0.003 0.027 0.098 0.088 0.069 0.012 0.0001l = 5 0.031 0.221 0.212 0.302 0.206 0.009 0.006 0.034 0.103 0.088 0.057 0.006 0.0001
Al,1|z| Al,2|z| Al,3|z| Al,4|z| Al,5|z| Al,6|z| Al,7|z| Al,8|z| Al,9|z| Al,10|z| Al,11
|z| Al,12|z| Al,13
|z|l = 1 0.342 0.671 0.759 0.405 0.163 0.042 0.013 0.248 0.243 0.243 0.412 0.199 0.042l = 2 0.335 0.453 0.621 0.352 0.972 0.043 0.022 0.336 0.297 0.223 0.332 0.25 0.04l = 3 0.494 0.579 0.316 0.567 0.309 0.041 0.016 0.216 0.225 0.235 0.359 0.197 0.049l = 4 0.338 0.348 0.764 0.319 0.152 0.033 0.024 0.144 0.264 0.223 0.327 0.226 0.042l = 5 0.138 0.595 0.637 0.238 0.142 0.032 0.024 0.218 0.273 0.222 0.308 0.184 0.042
Sl,13C Al,13
C
l = 1 0.004 0.022l = 2 0.001 0.019l = 3 0.003 0.016l = 4 0.001 0.025l = 5 0.001 0.025
S2,7L S3,7
L S4,7L S5,7
L A2,7L A3,7
L A4,7L A5,7
L
C = 7 0.006 0.009 0.012 0.01 0.027 0.018 0.027 0.029
Rolling Scheme Sl,1|z| Sl,2|z| Sl,3|z| Sl,4|z| Sl,5|z| Sl,6|z| Sl,7|z| Sl,8|z| Sl,9|z| Sl,10|z| Sl,11
|z| Sl,12|z| Sl,13
|z|
l = 1 0.528 0.46 0.843 0.472 0.142 0.002 0.001 0.007 0.008 0.093 0.281 0.026 0.0002l = 2 0.258 0.35 0.58 0.251 0.229 0.007 0.004 0.009 0.002 0.081 0.24 0.024 0.0003l = 3 0.559 0.31 0.741 0.437 0.16 0.002 0.001 0.002 0.002 0.066 0.288 0.013 0.0003l = 4 0.38 0.31 0.479 0.463 0.17 0.001 0.001 0.003 0.008 0.088 0.275 0.021 0.0002l = 5 0.285 0.249 0.297 0.225 0.12 0.007 0.001 0.003 0.003 0.063 0.327 0.019 0.0002
Al,1|z| Al,2|z| Al,3|z| Al,4|z| Al,5|z| Al,6|z| Al,7|z| Al,8|z| Al,9|z| Al,10|z| Al,11
|z| Al,12|z| Al,13
|z|l = 1 0.789 0.801 0.853 0.656 0.143 0.001 0.023 0.033 0.032 0.145 0.297 0.271 0.024l = 2 0.267 0.55 0.812 0.244 0.568 0.002 0.018 0.024 0.045 0.127 0.283 0.337 0.037l = 3 0.751 0.918 0.923 0.723 0.144 0.007 0.015 0.032 0.047 0.114 0.288 0.24 0.023l = 4 0.358 0.476 0.575 0.448 0.108 0.001 0.015 0.031 0.031 0.142 0.313 0.271 0.034l = 5 0.305 0.655 0.576 0.251 0.129 0.009 0.018 0.031 0.039 0.13 0.286 0.288 0.033
Sl,13C Al,13
C
l = 1 0.001 0.009l = 2 0.0003 0.004l = 3 0.0003 0.01l = 4 0.0003 0.005l = 5 0.002 0.004
S2,7L S3,7
L S4,7L S5,7
L A2,7L A3,7
L A4,7L A5,7
L
C = 7 0.008 0.008 0.006 0.009 0.038 0.025 0.019 0.031
Notes: 13 autocontours C = [0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99].
Sl,7|z| ,Al,7|z| for l = 1, 2, ...5; 7 refers to the 50% autocontour.
Sl,7L ,Al,7L stacking lags up to l = 2, ....5 and considering the 50% autocontour.
Sl,13C and Al,13
C stacking all 13 autocontours for one lag l = 1, 2, 3, 4, 5.500 bootstrap samples. T=649, R=360, P=289, m = 0.69
Table 14: Bootstrapped P-values: Linear Phillips Curve (Fixed and Rolling Schemes)
40
Model 3 Sl,1|z| Sl,2|z| Sl,3|z| Sl,4|z| Sl,5|z| Sl,6|z| Sl,7|z| Sl,8|z| Sl,9|z| Sl,10|z| Sl,11
|z| Sl,12|z| Sl,13
|z|
l = 1 0.452 0.662 0.454 0.406 0.466 0.694 0.702 0.814 0.788 0.57 0.41 0.254 0.296l = 2 0.428 0.552 0.218 0.464 0.436 0.678 0.652 0.816 0.684 0.55 0.41 0.276 0.296l = 3 0.476 0.784 0.454 0.424 0.466 0.66 0.602 0.874 0.782 0.582 0.43 0.206 0.296l = 4 0.656 0.738 0.676 0.55 0.584 0.72 0.807 0.884 0.82 0.584 0.386 0.224 0.296l = 5 0.652 0.538 0.422 0.444 0.536 0.78 0.836 0.844 0.842 0.578 0.416 0.244 0.296
Al,1|z| Al,2|z| Al,3|z| Al,4|z| Al,5|z| Al,6|z| Al,7|z| Al,8|z| Al,9|z| Al,10|z| Al,11
|z| Al,12|z| Al,13
|z|l = 1 0.442 0.644 0.43 0.39 0.568 0.682 0.856 0.822 0.818 0.596 0.45 0.278 0.294l = 2 0.402 0.446 0.318 0.362 0.326 0.488 0.674 0.834 0.72 0.57 0.446 0.308 0.296l = 3 0.606 0.748 0.446 0.388 0.454 0.558 0.804 0.868 0.79 0.606 0.434 0.23 0.294l = 4 0.638 0.768 0.666 0.348 0.58 0.722 0.86 0.874 0.824 0.584 0.4 0.226 0.296l = 5 0.632 0.514 0.404 0.432 0.522 0.722 0.83 0.852 0.86 0.582 0.412 0.246 0.39
Sl,13C Al,13
C
l = 1 0.614 0.596l = 2 0.48 0.454l = 3 0.494 0.482l = 4 0.51 0.478l = 5 0.488 0.472
S2,7L S3,7
L S4,7L S5,7
L A2,7L A3,7
L A4,7L A5,7
L
C = 7 0.518 0.566 0.548 0.598 0.534 0.578 0.564 0.618
Model 6 Sl,1|z| Sl,2|z| Sl,3|z| Sl,4|z| Sl,5|z| Sl,6|z| Sl,7|z| Sl,8|z| Sl,9|z| Sl,10|z| Sl,11
|z| Sl,12|z| Sl,13
|z|
l = 1 0.044 0.238 0.335 0.141 0.401 0.433 0.474 0.559 0.478 0.577 0.299 0.254 0.042l = 2 0.042 0.345 0.346 0.168 0.392 0.514 0.521 0.442 0.418 0.552 0.260 0.215 0.053l = 3 0.042 0.242 0.369 0.110 0.375 0.512 0.477 0.498 0.457 0.595 0.301 0.210 0.044l = 4 0.043 0.316 0.378 0.154 0.411 0.566 0.457 0.423 0.485 0.506 0.297 0.239 0.052l = 5 0.049 0.321 0.403 0.196 0.434 0.492 0.479 0.470 0.482 0.598 0.303 0.198 0.052
Al,1|z| Al,2|z| Al,3|z| Al,4|z| Al,5|z| Al,6|z| Al,7|z| Al,8|z| Al,9|z| Al,10|z| Al,11
|z| Al,12|z| Al,13
|z|l = 1 0.076 0.375 0.431 0.221 0.397 0.383 0.545 0.495 0.572 0.493 0.319 0.264 0.115l = 2 0.092 0.231 0.403 0.207 0.381 0.387 0.560 0.449 0.562 0.404 0.351 0.263 0.102l = 3 0.072 0.191 0.352 0.220 0.390 0.335 0.583 0.621 0.553 0.448 0.425 0.263 0.102l = 4 0.066 0.230 0.402 0.255 0.391 0.336 0.518 0.418 0.554 0.493 0.320 0.264 0.113l = 5 0.072 0.338 0.429 0.212 0.416 0.382 0.581 0.523 0.570 0.511 0.315 0.264 0.104
S13,lC A13,1
C
l = 1 0.366 0.480l = 2 0.299 0.433l = 3 0.323 0.449l = 4 0.326 0.436l = 5 0.351 0.446
S2,7C S3,7
C S4,7C S5,7
C A2,7C A3,7
C A4,7C A5,7
C
0.488 0.475 0.483 0.464 0.464 0.419 0.461 0.404
Notes: 13 autocontours C = [0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99].
Sl,7|z| ,Al,7|z| for l = 1, 2, ...5; 7 refers to the 50% autocontour.
Sl,7L ,Al,7L stacking lags up to l = 2, ....5 and considering the 50% autocontour.
Sl,13C and Al,13
C stacking all 13 autocontours for one lag l = 1, 2, 3, 4, 5.500 bootstrap samples.T=649, R=360, P=289, m = 0.69
Table 15: Bootstrapped P-values: Non-Linear Phillips Curve (Fixed Scheme)
41